What is a perfect square trinomial

Let’s first remember what a trinomial is. A polynomial has several terms. A trinomial (as the prefix “tri-” suggests) is a polynomial with three terms. When we’re dealing with perfect squares, it means we’re dealing with squaring binomials. Continue on to learn how we go about factoring a trinomial.

How to factor perfect square trinomials

One good way to recognize if a trinomial is perfect square is to look at its first and third term. If they are both squares, there’s a good chance that you may be working with a perfect square trinomial.

Let’s say we’re working with the following: x2+14x+49x^{2}+14x+49x​2​​+14x+49. Is this a perfect square trinomial? Looking at the first term, we’ve got x2x^{2}x​2​​, which is a square. The last term is 494949. It is also a square since when you multiply 777 by 777, you’ll get 494949. Therefore 494949 can also be written as 727^{2}7​2​​. The next step to identifying if we’ve got a perfect square is to see if we are able to get the middle term of 14x14x14x when we have x2x^{2}x​2​​and 727^{2}7​2​​ to work with.

In the case of a perfect square, the middle term is the first term multiplied by the last term, and then multiplied by 222. In other words, the perfect square trinomial formula is:

a2±ab+b2a^{2} \pm ab + b^{2}a​2​​±ab+b​2​​. We’re now trying to see if we can get the middle term of 2ab2ab2ab.

Since we’ve got our aaa term as xxx, and our bbb term as 777, our 2ab2ab2ab becomes 2∙7∙x2 \bullet 7 \bullet x2∙7∙x. That gives us a total of 14x14x14x, which is the middle term in x2+14x+49x^{2}+14x+49x​2​​+14x+49! Therefore, we can rewrite the question as (x+7)2(x + 7)^{2}(x+7)​2​​through factoring perfect square trinomials. You’ve solved a perfect square trinomial! You’re now ready to apply trinomial factoring to some practice problems.

Example problems

Question 1:

Factor the perfect square

x2−2x+36x^{2} - 2x + 36x​2​​−2x+36

Solution:

We know that this is a perfect square, and all we’re asked is to factor it. Therefore, just take a look at the first and last term and find what they are squares of. It’ll give us:

(x−6)2(x - 6)^{2}(x−6)​2​​

Question 2:

Factor the perfect square

3x2−30x+753x^{2} - 30x +753x​2​​−30x+75

Solution:

Take out the common factor 333

3(x2−10x+25)3(x^{2} - 10x + 25)3(x​2​​−10x+25)

Factor the x2−10x+25x^{2} - 10x + 25x​2​​−10x+25 and get the final answer:

3(x−5)23(x - 5)^{2}3(x−5)​2​​

Question 3:

Find the square of a binomial:

(−3x2+3y2)2(-3x^{2} + 3y^{2})^{2}(−3x​2​​+3y​2​​)​2​​

Solution:

You can square it and it will become what we have here:

ax2−bxy+cy2ax^{2} - bxy +cy^{2}ax​2​​−bxy+cy​2​​

So the first term:

Square of −3x2=9x4-3x^{2} = 9x^{4}−3x​2​​=9x​4​​

The third term:

3y2=9y43y^{2} = 9y^{4}3y​2​​=9y​4​​

The middle term is the multiplication of original 1st1^{st}1​st​​ and 2nd2^{nd}2​nd​​ term, and then times 222

To double check your answers, this online calculator will help you factor a polynomial expression. Use it as a reference, but make sure you learn how to properly go through the steps to answering a perfect square trinomial question.